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An almost complex manifolds $(M^4,J)$ of real dimension 4 with non-degeneratetorsion bundle admit a double absolute parallelism and it is provided theclassification of homogeneous $(M^4,J)$ having an associated non-solvable Liealgebra. We extend such a classification to the analysis of the manifoldshaving an associated solvable Lie algebra, up-to-coverings. Moreover, forhomogeneous $(M^4,J)$ we provide examples with connected and non-connecteddouble covering, thus proving that in general the double absolute parallelismis not the restriction of two absolute parallelisms. Furthermore, it is giventhe definition of a natural metric induced by the absolute parallelisms on$(M^4,J)$ and an example of an almost complex manifold with non-degeneratetorsion endowed with that metric such that it becomes an almost K\"ahlermanifold.

Coverings for $4$-dimensional almost complex manifolds with non-degenerate torsion